Improved upper bounds on the stabilizer rank of magic states

In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of $m$ copies of the magic state $|T\rangle=\sqrt{2}^{-1}(|0\rangle+e^{i\pi/4}|1\rangle)$ in the limit of large $m$. In particular, we show that $|T\rangle^{\otimes m}$ can be exactly expressed as a superposition of at most $O(2^{\alpha m})$ stabilizer states, where $\alpha\leq 0.3963$, improving on the best previously known bound $\alpha \leq 0.463$. This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an $n$-qubit Clifford + T circuit $U$ with $m$ uses of the T gate to within a given inverse polynomial relative error using a runtime $\mathrm{poly}(n,m)2^{\alpha m}$. We also provide improved upper bounds on the stabilizer rank of symmetric product states $|\psi\rangle^{\otimes m}$ more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and $m$ instances of any (fixed) single-qubit $Z$-rotation gate with runtime $\text{poly}(n,m) 2^{m/2}$. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.